On a problem of Gauss-Kuzmin type for continued fraction with odd partial quotients (Q1057385)

Let x be a number of the unit interval. Then x may be written in a unique way as a continued fraction \(x=1/(\alpha_ 1(x)+\epsilon_ 1(x)/(\alpha_ 2(x)+\epsilon_ 2(x)/(\alpha_ 3(x)+...))\) where \(\epsilon_ n\in \{-1,1\}\), \(\alpha_ n\geq 1\), \(\alpha_ n\equiv 1(mod 2)\) and \(\alpha_ n+\epsilon_ n>1\). Using the ergodic behaviour of a certain homogeneous random system with complete connections we solve a variant of Gauss-Kuzmin problem for the above expansion.